The above inequalities implies that if a ball covers a fraction of at least \(\varepsilon \)  points from \(P\)  (i.e., at least \(\varepsilon n\)  points), then it must cover at least one point of \(P\) . If we only ask for this (weaker) property from \(S\)  then \(S\)  is called an \(\varepsilon \) -net. To obtain the new results of this paper, we use a tool that generalizes the notion of \(\varepsilon \) -approximation and \(\varepsilon \) -net, but less common in the literature, and is known as \((\varepsilon ,\nu )\) -approximation [1]}.
